Instability of pole solutions for planar propagating flames in sufficiently large domains

It is well known that the partial differential equation (PDE) describing the dynamics of a hydrodynamically unstable planar flame front has exact pole solutions for which the PDE reduces to a set of ordinary differential equations (ODEs). The paradox, however, lies in the fact that the set of ODEs does not permit the appearance of new poles in the complex plane, or the formation of cusps in the physical space, as observed in experiments. The validity of the PDE itself has thus been questioned. We show here that the discrepancy between the PDE and the ODEs is due to the instability of exact pole solutions for the PDE. In previous work, we have reported that most exact pole solutions are indeed unstable for the PDE but, for each interval of relatively small length L, there remains one solution (up to translation symmetry) which is neutrally stable. The latter is a one-peak, coalescent solution for which the poles (whose number is maximal) are steady. The front undergoes bifurcations as the length of the domain considered increases: the one-pole, one-peak coalescent solution is first neutrally stable. As the length of the interval increases, it becomes unstable and the two-pole one-peak coalescent solution is, in turn, neutrally stable. This phenomenon occurs once again: as the two-pole solution becomes unstable, the three-pole solution becomes stable. The contribution of the present work is to show that subsequent bifurcations are of a different nature. As the interval length increases, the steady one-peak, coalescent solutions whose number of poles is maximal are no longer stable and bifurcations to unsteady states occur. In all cases, the appearance of new poles is observed in the unsteady dynamics. We also show analytically that such an instability is not permitted in the ODEs for which all steady one-peak, coalescent solutions are neutrally stable.